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Milne's rationality conjecture for abelian varieties

OpenAI · September 23, 2026

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This is an earlier edition (September 23, 2026). Current edition: Milne's rationality conjecture for abelian varieties, October 7, 2026.

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@misc{OAI:Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026,
  author = {{OpenAI}},
  title = {{Milne's rationality conjecture for abelian varieties}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026/paper.pdf}{OAI:Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026}},
  year = {2026}
}

Family 001: Milne’s rationality conjecture and algebraic specialization

Proves Milne's rationality conjecture for abelian varieties over $\overline{\mathbb Q}$ with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-p and crystalline realizations, for every residue characteristic p.

This is the only paper in this family.